# calculating work done by friction

I want to calculate the work done by friction if the length $L$ of uniform rope on the table slides off. There is friction between the cord and the table with coefficient of kinetic friction $\mu_k$.

$$W = \int F \cdot d \vec{s}$$ I think it would be: $$W_{fr} = \frac M L g \int_{0}^{L} dx$$

But the solutions (which could be mistaken) say: $$dW_{fr} = \mu_k \frac M L g \, x \, dx$$ which is then integrated.

Should there be an $x$ in the integral? I don't think there should be because you are summing up over an infinitesimal displacement $dx$ and the force of friction is not proportional to the displacement at any instant (I think).

• Do you know that your expression is dimensionally incorrect?
– ABC
Apr 6, 2013 at 3:27

Let $x$ denote the length of the rope that is on the table, then $$m(x) = \frac{M}{L}x$$ is the mass of the rope on the table. It follows that the force of friction on the rope on the table is $$f(x) = \mu_k m(x) g = \mu_k\frac{M}{L}xg$$ if the rope moves an amount $dx$ then the work done by friction is $$dW = f(x)dx = \mu_k\frac{M}{L}gx dx$$